# Lateral surface area

The ratio of the area of the base of the rotary cone to its lateral surface area is 3: 5. Calculate the surface and volume of the cone, if its height v = 4 cm.

Correct result:

S =  75.398 cm2
V =  37.699 cm3

#### Solution:

$v=4 \ \text{cm} \ \\ S_{1}:S_{2}=3:5 \ \\ \ \\ S_{1}=\pi r^2 \ \\ S_{2}=\pi r s \ \\ \ \\ 3/5=\dfrac{ \pi r^2 }{ \pi r s } \ \\ 3/5=r/s \ \\ \ \\ s^2=r^2 + v^2 \ \\ s^2=(3/5s)^2 + v^2 \ \\ \ \\ s=v / \sqrt{ 1 - (3/5)^{ 2 } }=4 / \sqrt{ 1 - (3/5)^{ 2 } }=5 \ \text{cm} \ \\ \ \\ r=3/5 \cdot \ s=3/5 \cdot \ 5=3 \ \text{cm} \ \\ \ \\ S_{1}=\pi \cdot \ r ^2=3.1416 \cdot \ 3 ^2 \doteq 28.2743 \ \text{cm}^2 \ \\ S_{2}=\pi \cdot \ r \cdot \ s=3.1416 \cdot \ 3 \cdot \ 5 \doteq 47.1239 \ \text{cm}^2 \ \\ \ \\ k=S_{1}/S_{2}=28.2743/47.1239=\dfrac{ 3 }{ 5 }=0.6 \ \\ \ \\ S=S_{1}+S_{2}=28.2743+47.1239=75.398 \ \text{cm}^2$
$V=\dfrac{ 1 }{ 3 } \cdot \ S_{1} \cdot \ v=\dfrac{ 1 }{ 3 } \cdot \ 28.2743 \cdot \ 4=37.699 \ \text{cm}^3$

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